Noether's theorem. One of the most important consequences of symmetries it that they gave rise to conserved quantities. This principle was first put forward by a German mathematician and theoretical physicist Emmy Nether. She discovered that the symmetries in the mathematical equations that described physical system automatically produce conserved quantities. The symmetry is preserved if a law remains invariant under transformations through space, translation through time and rotation about and axis.

You are properly familiar with conserved quantities like the laws of conservation of energy and momentum. The law of conservation of energy arise from the fact that equations are invariant with transitions in time-the total energy expenditure in a system of particles remains the same regardless of what time it is. Conservation of momentum is due tot he equations being invariant to translations in space. If two identical games of snooker are played which are separated in space but in which all the motions of the balls were the same, then you would expect the same player to be "snookered" whether the game was played in London or Edinburgh. These conservation laws of physics are closely connected with the properties of space and time and in general physicists believe them to be inviolable. From considerations of symmetry and using Noether's theorem, particle physicists have established a number of other conservation laws that enable to predict what particles will do when they interact with each other and what we can expect to see.

Consevation of Lepton number L. When we will discussed beta decay in Chapter 2, you will have noticed that in

decay, an antineutrino is emitted with the electron and never a neutrino, whereas in

decay, it is always a neutrino emitted with the positron. We assign each of these decays involving leptons a lepton number L. Leptons such as electrons muons tau lepton and neutrino have a lepton number of +1. Their antiparticles have a lepton number -1 whereas meason and baryons are assigned a lepton number of zero.

There are in fact three separate conservation laws for lepton number which correspond to the three varieties of lepton, e ,

and

To see how this works consider

decay:

The neutron and proton are baryons so they have L =0 and the electron e^- and antineutrino

, L = +1 and -1 respectively. You should see that the total lepton number (zero) in

decay remains the same for both sides of the decay. When we are observing leptons we must make a note of each type of lepton ( e ,

and

separately. There is a distinction between electron and muno type leptons which is shown by the following processes. Experiments with particle accelerators have been carried out where beams of muon type antineutrinos

are incident on a proton target and the following reaction takes place

Now if there was no difference between elcectron and muon type leptons (i.e ,

e+ ) then the following reaction should also be possible

However this reaction is never observed. This indicates that there must be fundamental differences between the type of lepton and we need to take this into account when we calculate the total lepton number of particle reaction. We give the symbols

,

and

to these separate lepton numbers for the different families of leptons.

We state the Conservation of Lepton Number as In processes involving particle reactions the lepton number for electron-type leptons, muno type leptons and tau-type leptons, must remain the same

By imposing the constraint that lepton number must be conserved in particle reactions, particle physicists can explain why neutrinos and sometimes antineutrinos (or sometimes both in the case of muno decay) must be observed.

Conservation of Bayron number

We can state anther conservation law involving baryons. We assign a baryon number B of B = +1 and baryon number of -1 to antibaryons with all non-baryons (mesons and leptons having B = 0) We state the law of Conservation of Baryon Numbers as

In any aprticle reaction the total baryon number remains the same

the conservation of neutron number A is a special case of conservation of baryon number in which all the baryons happens to be either protons or neutrons. While nuclear physicists use the symbol A to represent nucleons, particle physicists always use the symbol B to represent all baryons including nucleons. For example, the antiproton

was discovered by means of the following particle reaction

We can confirm that baryon number is conserved.

The totoal baryon number is +2 on both sides therefore B is conserved and the reaction is possible.

Conservation of Strangeness

Recall that strangeness was introduced to explain why some hadrons like the kaon (K) and the lambda (

) had unexpectedly long lifetimes when interacting with the strong force. All particles are either strange or non-strange. In kaon decay, the K⁰ and the K⁺ are assigned strangeness S = +1, K^- has S = -1 , pions and leptons are non-strange particles and have S = 0. Strangeness can be used to explain the properties of certain particle decays. Particle physicists have found that strangeness is always conserved in interactions involving strong force or electromagnetic forces but in the weak interaction the strangeness is either conserved or changes by +1. We can state this as the law of conmservation of strangeness

For particle interactions governed by the strong or electromagnetic forces, the total strangeness must remain the same. For interactions governed by the weak force, the strangeness either remains the same or changes by one unit.

Strangeness can be explained why strange particles are always produced in pairs. Pions are produced in nuclear collisions and since there is no "meson conservation law", in theory any number could be produced. But the K mesons and other strange particles are always produced in pairs. For example for any general K-meson we could have

Since the proton is a non-strange particle with S = 0, this reaction could be explained if the first K meson in this reaction is a K⁺ or K⁰ with S = +1 and the other is a K^- or

with S = -1. Similarly the reaction

can be explained if the

has S = -1. If we do this, we see that the total strangeness number (in this case zero) is the same on both sides and is therefore conserved. This phenomenon, where strange particles are produced in pairs, is called associated production . Conservation of strangeness also tells us what force particles interact with and which reactions

is a strange particle that is observed to decay by the process

with a lifetime of about 10^-10s. But being a baryon we would expect this strongly interacting particle to decay to other strongly interacting particles with characteristic life times of 10^-23s. However if we assign S = -1 to the

, strangeness is not conserved ( the p and

are non-strange ) and so the reaction cannot go by the strong interaction. It must therefore be due to the weak interaction which gives its reactively long lifetime of 10^-10s. Do not be mystified by the use of the term "strangeness" to describe particles. There is nothing inherently strange about them in the intuitive sense. They were only called strange because their behaviour was initially unexpected and strangeness is simply a quantum property of particles just like a charge or a spin. Particle physicists do not know what lepton number, baryon number and strangeness really mean. They are merely useful concepts that emerged by considering mathematical symmetries and their associated conservation properties which enables us to describe the ways in which particles can decay into other particles. The name' strangeness' stuck for historical reasons however, and we will see later that particle physicists have fondness for labelling properties of particles with whimsical names!